课程： 11.8 对数属性 | The Way To Learn

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对对数属性
The exposure value or value for a "correct exposure" can be calculated using the formula $\begin{aligned} EV = \log_{2} (\frac{N^{2}}{t}), \end{aligned}$ where EV is the exposure value, $\begin{aligned} N \end{aligned}$ is the aperture or f-stop, and $\begin{aligned} t \end{aligned}$ is time in seconds ¹ . (The image below demonstrates pictures of the same image using different exposure values.) We can make this easier to calculate if one of the quantities is fixed by u sing properties of logarithms. We describe those properties in this section.
::“ 正确暴露” 的暴露值或值可以用 EV=log2(N2t) 公式计算,EV 是暴露值, N 是孔径或 f- stop , t 是秒数 1 。 (下图用不同曝光值来显示相同图像的图片。) 如果使用对数的属性来固定其中的一个数量, 我们可以比较容易地计算。我们在本节描述这些属性。

Logarithmic Properties
::对对数属性

Previously, we have discussed the exponential properties. Let's recall them below.
::之前我们讨论过指数性,下面回顾一下

Exponential Properties
::指数属性

$\begin{aligned} b^{0} = 1 & Zero Exponent \\ b^{1} = b & Identity \\ b^{m} \cdot b^{n} = b^{m + n} & Product Rule \\ \frac{b^{m}}{b^{n}} = b^{m - n} & Quotient Rule \\ (b^{m})^{n} = b^{m n} & Power Rule \end{aligned}$

Each of these has analogous logarithmic properties.
::其中每一个都有类似的对数属性。

Zero Exponent and Identity Properties
::零指数和身份属性

Zero Exponent and Identity Properties of Logarithms
::对数零指数和特性属性

Zero Exponent Property
::零指数属性

$\begin{aligned} \log_{b} 1 = 0 \end{aligned}$
Identity Property
::logb1=0身份属性

$\begin{aligned} \log_{b} b = 1 \end{aligned}$

::对数bb=1

Example 1
::例1

Evaluate the following logarithms:
::评估下列对数:

a. $\begin{aligned} \log_{8} 1 \end{aligned}$
::a. 日对数81

b. $\begin{aligned} \log_{π} 1 \end{aligned}$
::b. log%1

Solution: For both of these logarithms, we can use the zero exponent property above. Both are equal to zero.
::解决方案 : 对于这两个对数, 我们可以使用上面的零引号属性。两者均等于零。

Another way to think about these is to express them in exponential form.
::另一种思考这些的办法是以指数形式表达它们。

$\begin{aligned} 8^{?} = 1 & π^{?} = 1 \end{aligned}$
In both of these cases, the exponent that works is zero.
::8? =1? =1 在这两种情况下, 功用指数为零。

Example 2
::例2

Evaluate the following logarithms:
::评估下列对数:

a. $\begin{aligned} \log_{7} 7 \end{aligned}$
::a. log7+%7

b. $\begin{aligned} \log_{\sqrt{2}} \sqrt{2} \end{aligned}$
::b. log2+%2

Solution: The identity property can be used to evaluate both of these logarithms. They are each equal to 1.
::解答:身份属性可用于评估这两个对数,每个对数等于1。

Again, if we convert these to exponential form, we can see that the exponent 1 works.
::再说一遍,如果我们把这些转换成指数形式, 我们可以看到推力1起作用了。

$\begin{aligned} 7^{?} = 7 & {\sqrt{2}}^{?} = \sqrt{2} \end{aligned}$

Product Rule of Logarithms
::产品对数规则

The product rule of exponents has a corresponding rule: the product rule of logarithms.
::指数者的产品规则有相应的规则:对数的产品规则。

Product Rule of Logarithms
::产品对数规则

$\begin{aligned} \log_{b} (x y) = \log_{b} x + \log_{b} y \end{aligned}$

First, notice that these logs have the same base. If they do not, then this property does not apply.
::首先, 请注意这些日志有相同的基数。如果它们没有, 那么此属性不适用。

Let's derive the product rule of logarithms using the product rule of exponents. Start with $\begin{aligned} \log_{b} x + \log_{b} y \end{aligned}$ .
::让我们使用引言方的产物规则来得出对数的产物规则。从logbx+logby开始。

Set $\begin{aligned} \log_{b} x = m \end{aligned}$ and $\begin{aligned} \log_{b} y = n \end{aligned}$ so the right side of the product rule is equal to $\begin{aligned} m + n \end{aligned}$ .
::设置对数bx=m和对数by=n,产品规则的右侧等于 m+n。

For the left side, convert to exponential form $\begin{aligned} b^{m} = x \end{aligned}$ and $\begin{aligned} b^{n} = y \end{aligned}$ , and multiply these together.
::左侧,转换成指数形式 bm=x 和 bn=y,并同时乘以这些。

$\begin{aligned} b^{m} \cdot b^{n} & = x y \\ b^{m + n} & = x y \end{aligned}$

::bmbn=xybm+n=xy

Now, convert back to logarithmic form to get the left side of the product rule.
::现在, 转换为对数表, 以获得产品规则的左侧。

$\begin{aligned} b^{m + n} = x y ⟹ \log_{b} x y = m + n \end{aligned}$

::bm+n=xylogbxy=m+n

The two sides of this equation are equal to $\begin{aligned} m + n \end{aligned}$ . Therefore, $\begin{aligned} \log_{b} x y = \log_{b} x + \log_{b} y . \end{aligned}$
::此方程式的两面等于 m+n。因此, logbxy=logbx+logby 。

Example 3
::例3

Expand $\begin{aligned} \log_{12} 4 y \end{aligned}$ .
::展开对数124y。

Solution: The argument is a product of $\begin{aligned} 4 \end{aligned}$ and $\begin{aligned} y \end{aligned}$ . Using the product rule, we have
::解决办法:这一论点是4和y的产物。

$\begin{aligned} \log_{12} 4 y = \log_{12} 4 + \log_{12} y . \end{aligned}$

::log124y=log124+log12y。

by Mathispower4u demonstrates how to use the product rule of logarithms to expand a logarithm.
::Mathispower4u 演示如何使用对数的产品规则来扩展对数。

WARNING
::警告

The order in the product rule is important. It is the log of a product and not the product of logs. The following is incorrect:
::产品规则中的顺序很重要。它是一个产品的日志, 而不是日志的产物。以下不正确 :

$\begin{aligned} (\log_{b} x) (\log_{b} n) \neq \log_{b} x + \log_{b} y . \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(logbx(logbn)) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(logbx+logby) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}(logbx) {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

Also, you cannot distribute the "log" to each of the terms you are taking the log of. The following is incorrect:
::此外,您不能将“ log” 分布到您所选的每个术语中。以下不正确 :

$\begin{aligned} \log_{b} (x + y) \neq \log_{b} x + \log_{b} y . \end{aligned}$

::logb(x+y) logbx+logby 。

Quotient Rule of Logarithms
::引引对数规则

The quotient rule of exponents also has a rule that follows from it in logarithmic form.
::引言人的商数规则也有一条以对数形式遵循的规则。

Quotient Rule of Logarithms
::引引对数规则

$\begin{aligned} \log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y \end{aligned}$

We can prove this using the quotient rule of exponents. We start with the expression $\begin{aligned} \log_{b} x - \log_{b} y, \end{aligned}$ and define $\begin{aligned} \log_{b} x = m \end{aligned}$ and $\begin{aligned} \log_{b} y = n \end{aligned}$ . The right-hand side of the quotient rule of logarithms equals $\begin{aligned} m - n \end{aligned}$ .
::我们可以使用引数规则来证明这一点。我们从表达式 logbx- logby 开始, 定义logbx=m 和 logby=n。对数规则的右手侧等于 m-n 。

To show the left-hand side also equals $\begin{aligned} m - n \end{aligned}$ , we convert our definitions to exponential form: $\begin{aligned} b^{m} = x \end{aligned}$ and $\begin{aligned} b^{n} = y \end{aligned}$ . Now,
::要显示左手侧也等于 m- n, 我们将把定义转换成指数形式: bm=x 和 bn=y。现在,

$\begin{aligned} \frac{x}{y} = \frac{b^{m}}{b^{n}} = b^{m - n} \end{aligned}$
The $\begin{aligned} \log_{b} (\frac{x}{y}) = \log_{b} (b^{m - n}) = m - n . \end{aligned}$
::xy=bmbn=bm-n-logb(xy)=logb(bm-n)=m-n。

Since both sides are equal to $\begin{aligned} m - n \end{aligned}$ , the quotient rule of logarithms holds.
::由于双方均等于 m-n,对数的商数规则维持不变。

Example 4
::例4

Simplify $\begin{aligned} \log_{3} 15 - \log_{3} 5. \end{aligned}$
::简化日志3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Solution: We are subtracting two logs with the same base, so we can use the quotient rule:
::解答:我们正在用相同的基数减去两个日志, 所以我们可以使用商数规则:

$\begin{aligned} \log_{3} 15 - \log_{3} 5 & = \log_{3} \frac{15}{5} \\ = \log_{3} 3 \\ = 1 \end{aligned}$

::对数 3\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\8\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 5
::例5

Expand $\begin{aligned} \log_{2} (\frac{32}{z}) \end{aligned}$ .
::展开对数2( 32z) 。

Solution: We can use the quotient rule of logarithms to write this as two separate logarithms.
::解答: 我们可以使用对数的商数规则将此写成两个独立的对数。

$\begin{aligned} \log_{2} (\frac{32}{z}) = \log_{2} 32 - \log_{2} z = 5 - \log_{2} z \end{aligned}$

::log2(32z) = log2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

by CK-12 shows how to evaluate a logarithm using the quotient rule.
::使用 CK-12 显示如何使用商数规则对对数进行评价。

WARNING
::警告

The order is important. The quotient rule is the log of a quotient, not the quotient of logs. The following is incorrect:
::顺序很重要。商数规则是商数的日志, 而不是日志的商数。以下不正确 :

$\begin{aligned} \frac{\log_{b} x}{\log_{b} y} \neq \log_{b} x - \log_{b} y \end{aligned}$
If you have a quotient of logs, you cannot cancel the "log." The following is incorrect:
::logbxlogbylogbx-logby 如果您有日志的商数, 您无法取消“ log” 。以下不正确 :

$\begin{aligned} \frac{\log_{b} x}{\log_{b} y} \neq \frac{x}{y} \end{aligned}$
"Log" never operates by itself. It must be followed by a number or a variable or an expression.
::logbxlogbyxy "Log" 绝非自动操作。之后必须有一个数字或变量或表达式。

Power Rule of Logarithms
::对数规则的权力

Just like the other rules of exponents, the power rule of exponents has a related power rule of logarithms.
::就像其他推手的规则一样推手的权力规则也有相关的对数规则

Power Rule of Logarithms
::对数规则的权力

$\begin{aligned} \log_{b} x^{n} = n \log_{b} x \end{aligned}$

To show this rule is correct, set $\begin{aligned} \log_{b} x = y . \end{aligned}$
::要显示此规则正确, 请设定logbx=y 。

Converting to exponential form , we have $\begin{aligned} b^{y} = x \end{aligned}$ . Now, raise both sides to the $\begin{aligned} n \end{aligned}$ power.
::正在转换成指数形式, 我们有 by=x 。现在, 将两边都升到 n 电源。

$\begin{aligned} (b^{y})^{n} & = x^{n} \\ b^{n y} & = x^{n} \end{aligned}$

:n)n=xnbny=xn)

Let's convert this back to a log with base $\begin{aligned} b \end{aligned}$ , $\begin{aligned} \log_{b} x^{n} = \log_{b} b^{n y} = n y . \end{aligned}$ Substituting for $\begin{aligned} y \end{aligned}$ , we have $\begin{aligned} \log_{b} x^{n} = n \log_{b} x . \end{aligned}$
::将它转换为以 b, logbxn=logbbny=ny为基准的日志。替代 y, 我们有logbxn=nlogbx。

Example 6
::例6

Simplify $\begin{aligned} \log_{9} 81^{x + 2} \end{aligned}$ .
::简化对数 9\\\\\\ 81x+2 。

Solution: W e can move the power in front of the logarithm by using the power rule:
::解答:我们可以使用权力规则在对数前面移动权力:

$\begin{aligned} \log_{9} 81^{x + 2} & = \log_{9} 9^{2 (x + 2)} \\ = 2 (x + 2) \\ = 2 x + 4 \end{aligned}$

::对数 9\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\4

Example 7
::例7

Find $\begin{aligned} \log_{4} 16^{x} . \end{aligned}$
::查找log416x。

Solution: R ewrite 16 as $\begin{aligned} 4^{2} . \end{aligned}$
::解决办法:重写16为42。

$\begin{aligned} \log_{4} 16^{x} = \log_{4} (4^{2})^{x} = \log_{4} 4^{2 x} = 2 x \end{aligned}$

::log4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 8
::例8

Expand $\begin{aligned} \log (5 c^{4})^{2} \end{aligned}$ .
::展开对数(5c4) 2。

Solution: Here , you will need to apply the power rule twice and the product rule.
::解决方案:在这里,你需要两次应用权力规则和产品规则。

$\begin{aligned} \log (5 c^{4})^{2} & = 2 \log 5 c^{4} & power rule \\ = 2 (\log 5 + \log c^{4}) & product rule \\ = 2 (\log 5 + 4 \log c) & power rule \\ = 2 \log 5 + 8 \log c \end{aligned}$

::log=2log_5c4 power rule=2(log_5+log_c4) 产品规则=2(log_5+4log}c) 电源规则=2log_5+8log}c

WARNING
::警告

There are several different ways to express one logarithm. For example, the following are all equivalent logarithms:
$\begin{aligned} \log (5 c^{4})^{2} = \log 25 + 8 \log c = \log 25 + \log c^{8} = \log 25 c^{8} \end{aligned}$
Notice we did not move the exponent of 8 in front of both logarithms. We could not do this because it applied only to the variable $\begin{aligned} c . \end{aligned}$
::表示一个对数有几种不同的方式。例如, 以下是所有等效对数 : log( 5c4) 2 = log 25+8logc=log25+logc8=log25+logc8=log25c8Notice 我们没有在两个对数前移动 8 的对数。我们无法这样做, 因为它只适用于变量 c 。

by B Columbia Calculus shows several examples of applying the power rule.
::B Columbia Calculus在应用权力规则时举出了几个例子。

WARNING
::警告

Again, the order is important. The power rule is the log of a result raised to a power, not a log raised to a power. The following is incorrect:
::同样,命令是重要的。权力规则是向权力提出结果的日志, 而不是向权力提出结果的日志。以下不正确 :

$\begin{aligned} (\log_{b} x)^{n} \neq n \log_{b} x . \end{aligned}$

:logbx)nnlogbx。

Example 9
::例9

The exposure value or value for a "correct exposure" can be calculated using the formula, $\begin{aligned} EV = \log_{2} (\frac{N^{2}}{t}), \end{aligned}$ where EV is the exposure value, $\begin{aligned} N \end{aligned}$ is the aperture or f-stop, and $\begin{aligned} t \end{aligned}$ is time in seconds ¹ . We can write this in expanded form, which would make repeated calculations of the time needed for a correct exposure given a particular EV and f-stop easier. Write this formula in expanded form.
::“ 正确暴露” 的暴露值或值可以使用 EV=log2(N2t) 公式计算,EV 是暴露值, N 是孔径或 f- stop, t 是秒1 的时间。我们可以以扩展的形式写下这个公式, 这样可以根据特定 EV 和 f- stop 的方便度, 重复计算正确暴露所需的时间。以扩展的形式写入此公式。

$\begin{aligned} EV & = \log_{2} (\frac{N^{2}}{t}) \\ = \log_{2} N^{2} - \log_{2} t & quotient rule \\ = 2 \log_{2} N - \log_{2} t & power rule \end{aligned}$

::EV = log2 (N2t) = log2 N2 - log2 tquotient rule= 2log2N- log2t power rule

Inverse Functions
::反反函数

We have continued to use the inverse relationship between exponential and in this section. We can define that relationship with a property.
::我们继续使用指数和本节中的反比关系,我们可以界定这种关系与财产的关系。

Inverse Property of Logarithms
::对数的反对数属性

$\begin{aligned} b^{\log_{b} x} = x \end{aligned}$

::博客bx=x

We have not had a logarithm in the exponent yet, but, remember, that is what logarithms represent. We can see that this property is true if we define $\begin{aligned} f (x) = b^{x} \end{aligned}$ and $\begin{aligned} g (x) = \log_{b} x \end{aligned}$ . These two functions are inverses of each other by definition of the logarithm. When two inverses are composed, they equal $\begin{aligned} x \end{aligned}$ . We did this as a check in the section.
::我们还没有在对数表里有一个对数, 但记住, 这就是对数代表的对数。我们可以看到, 如果我们定义 f( x) =bx 和 g( x) =logb* x, 这个属性是真实的。这两个函数根据对数表的定义是互相反相的。当两个对数组成时, 它们等于 x。我们在区域中检查了这个属性。

The inverse property is essentially $\begin{aligned} (f \circ g) (x) = f (g (x)) = f (\log_{b} x) = b^{\log_{b} x} . \end{aligned}$ Since these two functions are inverses, this equals $\begin{aligned} x \end{aligned}$ .
::反向属性基本上为 (fg)(x) =f(g(x)) =f(logbx) =blogbx。由于这两个函数是反向的,此等值等于 x。

Example 10
::例10

Evaluate the following:
::评估以下方面:

a. $\begin{aligned} 10^{\log 56} \end{aligned}$
::a. 10log*56

b. $\begin{aligned} e^{\ln 6} \cdot e^{\ln 2} \end{aligned}$
::b. 2

c. $\begin{aligned} 5^{\log_{5} 6 x} \end{aligned}$
::c. 5log56x

Solution:
::解决方案 :

a. Since the base is 10 for the exponential part and the logarithm, we can use the inverse property: $\begin{aligned} 10^{\log 56} = 56 \end{aligned}$
::a. 由于指数部分和对数的基数是10,因此我们可以使用反向属性:10log56=56

b. Here, $\begin{aligned} e \end{aligned}$ and the natural log have the same base. After using the inverse property, we are left with $\begin{aligned} 6 \cdot 2 = 12 \end{aligned}$ .
::b. 这里,e 和自然日志的基数相同。在使用反向属性后,我们只剩下62=12。

c. The bases are the same, so with the inverse property, $\begin{aligned} 5^{\log_{5} 6 x} = 6 x \end{aligned}$ .
::c. 基数相同,反向属性为5log56x=6x。

We can also use this property to find inverse functions.
::我们也可以使用此属性查找反函数。

Example 11
::例例11

Find the inverse of $\begin{aligned} f (x) = \log (x + 3) . \end{aligned}$
::查找 f( x) =log( x+3) 的反方向。

Solution: We will follow the same steps we use to find inverse functions.
::解决方案:我们将遵循我们用来查找反函数的相同步骤。

$\begin{aligned} f (x) & = \log (x + 3) \\ y & = \log (x + 3) & replace f (x) with y \\ x & = \log (y + 3) & switch x and y \end{aligned}$

::f( x) =log}( x+3) y=log}( x+3) y=log}( x+3) =log}( x+3) =log} (x) 替换 f( x) 与 Yx=log} (y+3) switch x 和 Y

We need to get the $\begin{aligned} y \end{aligned}$ out of the logarithm. If we think of each side of this equation as an exponent, if we raised the same base to each of these powers, we should get the same result since they are equal. This will allow us to use the inverse property to get $\begin{aligned} y \end{aligned}$ outside of the logarithm and find the inverse function.
::我们需要把y 调出对数。如果我们把方程式的每个侧面看成一个引号, 如果我们把这些权力的每一个都提升到相同的基点, 我们应该得到相同的结果, 因为它们是平等的。这将允许我们使用反向属性将y 带出对数, 并找到反向函数。

$\begin{aligned} 10^{x} & = 10^{\log (y + 3)} \\ 10^{x} & = y + 3 & inverse property \\ 10^{x} - 3 & = y \\ 10^{x} - 3 & = f^{- 1} (x) & replace y with f^{- 1} (x) \end{aligned}$

::10x=10log(y+3)10x=y+3inverseal10x-3=y10x-3=y10x-3=f-1(x)replace y with f-1(x)

Example 12
::例例12

Find the inverse of $\begin{aligned} y = - \ln x + 2 \end{aligned}$ .
::查找 y=- lnx+2 的反白。

Solution: Following our steps and using the inverse property, we have
::解决办法:按照我们的步骤,使用反向财产,

$\begin{aligned} y & = - \ln x + 2 \\ x & = - \ln y + 2 \\ x - 2 & = - \ln y \\ 2 - x & = \ln y \\ e^{2 - x} & = e^{\ln y} \\ e^{2 - x} & = y \\ y^{- 1} & = e^{2 - x} \end{aligned}$

::y= y= y= y= y= e2-x y= yy= yy= yy= y2=- ln= y2=- ln= y2=x= ln= ye2-x= eln= eln= ye2-x=yyy=e2-x

by The Math Sorcerer demonstrates how to find the inverse of a logarithmic function.
::的数学搜索器演示了如何找到对数函数的反函数。

Example 13
::例13

Find the inverse of $\begin{aligned} f (x) = 2 e^{x - 1} \end{aligned}$ .
::查找 f( x) =2ex- 1 的反义。

Solution: Change $\begin{aligned} f (x) \end{aligned}$ to $\begin{aligned} y \end{aligned}$ . Then, switch $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::解析度: 将 f(x) 更改为 y。然后, 切换 x 和 y 。

$\begin{aligned} y = 2 e^{x - 1} \\ x = 2 e^{y - 1} \end{aligned}$

::y=2ex- 1x=2ey- 1

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.
::现在,我们需要孤立推手并取出双方的对数。第一除以2。

$\begin{aligned} \frac{x}{2} = e^{y - 1} \\ \ln (\frac{x}{2}) = \ln e^{y - 1} \end{aligned}$

::x2=ey- 1ln_( x2)=ln_ ey- 1

For t he right side of our equation, we can use the power rule and the identity property to get $\begin{aligned} \ln e^{y - 1} = y - 1 \end{aligned}$ . Solve for $\begin{aligned} y \end{aligned}$ .
::对于我们等式的右侧, 我们可以使用权力规则和身份属性来获得 iney -1=y -1. 解决y。

$\begin{aligned} \ln (\frac{x}{2}) = y - 1 \\ \ln (\frac{x}{2}) + 1 = y \end{aligned}$

::In(x2) =y - 1ln(x2)+1=y

Therefore, $\begin{aligned} f^{- 1} (x) = \ln (\frac{x}{2}) + 1 \end{aligned}$ is the inverse of $\begin{aligned} f (x) = 2 e^{x - 1} \end{aligned}$ .
::因此, f-1(x) = ln (x2)+1 是 f(x) = 2ex-1 的逆数。

Example 14
::例14

Find the inverse of $\begin{aligned} f (x) = 4^{x + 2} - 5 \end{aligned}$ .
::查找 f( x) =4x+2 - 5 的反义。

Solution: Follow the steps to find the inverse.
::解决方案:按照步骤寻找反向。

$\begin{aligned} f (x) & = 4^{x + 2} - 5 \\ y & = 4^{x + 2} - 5 \\ x & = 4^{y + 2} - 5 \\ x + 5 & = 4^{y + 2} \\ \log_{4} (x + 5) & = y + 2 \\ \log_{4} (x + 5) - 2 & = y \end{aligned}$

::f( x) = 4x+2-2- 5y= 4x+2-2-5x= 4y+2-5x+5= 4y+2log4( x+5) =y+2log4( x+5) - 2=y

by R Parsons demonstrates how to find the inverse of an exponential function.
::R Parsons 展示了如何找到指数函数的反向函数。

Summary
::摘要

The logarithmic properties are:

Zero Exponent: $\begin{aligned} \log_{b} 1 = 0 \end{aligned}$
::零指数: logb1=0

Identity: $\begin{aligned} \log_{b} b = 1 \end{aligned}$
::身份: logbb=1

Product Rule: $\begin{aligned} \log_{b} (x y) = \log_{b} x + \log_{b} y \end{aligned}$
::产品规则: logb( xy) =logbx+logby

Quotient Rule: $\begin{aligned} \log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y \end{aligned}$
::引号规则: logb(xy) =logbx-logby

Power Rule: $\begin{aligned} \log_{b} x^{n} = n \log_{b} x \end{aligned}$
::规则 : logbxn=nlogbx

Inverse: $\begin{aligned} b^{\log_{b} x} = x \end{aligned}$
::倒数: 博客bx=x

::对数属性为: 零指数: logb1=0 身份: logbb=1 产品规则: logb(xy) = logbx+logby Quotient 规则: logb(xy) = logbx-logby 动力规则: logbxn=nlogbx Inverse: 博客bx=x

To find the inverse of a logarithmic function, you can use the inverse property or convert to exponential form.
::要查找对数函数的反向,您可以使用反向属性或转换成指数形式。

To find the inverse of an exponential function, you need to take the logarithm of both sides.
::要找到指数函数的逆函数, 您需要使用两侧的对数。

Review
::回顾

Simplify the following expressions:
::简化以下表达式:

1. $\begin{aligned} \log_{3} 27^{x} \end{aligned}$
::1. 日数327x

2. $\begin{aligned} \log_{5} {(\frac{1}{5})}^{x} \end{aligned}$
::2. log5(15)x

3. $\begin{aligned} 10^{\log (x + 3)} \end{aligned}$
::3. 10log(x+3)

4. $\begin{aligned} \log_{6} 36^{(x - 1)} \end{aligned}$
::4. log636(x-1)

5. $\begin{aligned} e^{\ln (x - 7)} \end{aligned}$
::5. (x-7)

6. $\begin{aligned} \log {(\frac{1}{100})}^{3 x} \end{aligned}$
::6. 对数(1100)3x

7. $\begin{aligned} \log_{7} y^{2} \end{aligned}$
::7. log7y2

8. $\begin{aligned} \log_{4} (9 x)^{3} \end{aligned}$
::8. log4(9x)3

9. $\begin{aligned} \ln 7 - \ln 14 + \ln 10 \end{aligned}$
::9. 内 - 内 - 内 - 内 - 十四+一内 - 十

10. $\begin{aligned} \log_{11} 22 + \log_{11} 5 - \log_{11} 55 \end{aligned}$
::log11+log11_5_log11_55

Expand the following expressions:
::展开以下表达式 :

11. $\begin{aligned} \log_{12} 5 z^{2} \end{aligned}$
::11.125z2对数

12. $\begin{aligned} \log_{6} (5 x) \end{aligned}$
::12.6(5x)对数

13. $\begin{aligned} \log_{3} (a b c) \end{aligned}$
::13.3(abc)对数

14. $\begin{aligned} \log_{9} (\frac{x y}{5}) \end{aligned}$
::14.9(xy5)

15. $\begin{aligned} \log (\frac{2 x}{y}) \end{aligned}$
::15. 对数(2xy)

16. $\begin{aligned} \log_{4} (\frac{5}{9 y}) \end{aligned}$
::16.4(59y)

Find the inverse of each of the following exponential functions:
::查找下列指数函数的反向 :

17. $\begin{aligned} y = 3 e^{x + 2} \end{aligned}$
::17.y=3ex+2

18. $\begin{aligned} f (x) = \frac{1}{5} e^{\frac{x}{7}} \end{aligned}$
::18. f(x)=15ex7

19. $\begin{aligned} y = 2 + e^{2 x - 3} \end{aligned}$
::19. y=2+e2x-3

20. $\begin{aligned} g (x) = \log_{5} (3 x - 1) \end{aligned}$
::20. g(x) = log5 (3x- 1)

21. $\begin{aligned} y = \log (- \frac{x}{2}) \end{aligned}$
::21.y=log(-x2)

22. $\begin{aligned} y = \ln (4 x) - 2 \end{aligned}$
::y= y y= ln ( 4x) - 2

Explore More
::探索更多

1. The hypotenuse of a right triangle has a length of $\begin{aligned} \log_{3} 27^{8} . \end{aligned}$ How long is the triangle's hypotenuse?
::1. 右三角形的下限长度为log3 278. 三角形的下限持续多久?

2. Write an example for each of the warnings in this section that demonstrates that it is incorrect.
::2. 为本节中的每个警告写一个实例,说明其不正确。

3. Are $\begin{aligned} y = \log x^{2} \end{aligned}$ and $\begin{aligned} y = 2 \log x \end{aligned}$ the same function? Explain your reasoning.
::3. y=logx2和y=2logx是否相同功能?解释一下你的推理。

4. What is wrong with the following?
::4. 以下各点有什么问题?

$\begin{aligned} \log (\frac{1}{2}) & < 3 \log (\frac{1}{2}) \\ \log (\frac{1}{2}) & < \log (\frac{1}{2})^{3} \\ \log (\frac{1}{2}) & < \log (\frac{1}{8}) \\ \frac{1}{2} & < \frac{1}{8} \end{aligned}$

::log_(12)<(12)<(12)>(12)<(12)><(12)_(12)_(12)_(12)_(12)_(12)_(18)_(18)_(12)_(18)_

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try this interactive that reinforces the concepts explored in this section:
::尝试这一互动,强化本节所探讨的概念:

References
::参考参考资料

1. "Exposure Value," last edited May 19, 2017,
::1. 2017年5月19日上次编辑的《曝光值》

章节大纲

常规

对对数属性

Logarithmic Properties ::对对数属性

Exponential Properties ::指数属性

Zero Exponent and Identity Properties ::零指数和身份属性

Zero Exponent and Identity Properties of Logarithms ::对数零指数和特性属性

Example 1 ::例1

Example 2 ::例2

Product Rule of Logarithms ::产品对数规则

Product Rule of Logarithms ::产品对数规则

Example 3 ::例3

WARNING ::警告

Quotient Rule of Logarithms ::引引对数规则

Quotient Rule of Logarithms ::引引对数规则

Example 4 ::例4

Example 5 ::例5

WARNING ::警告

Power Rule of Logarithms ::对数规则的权力

Power Rule of Logarithms ::对数规则的权力

Example 6 ::例6

Example 7 ::例7

Example 8 ::例8

WARNING ::警告

WARNING ::警告

Example 9 ::例9

Inverse Functions ::反反函数

Inverse Property of Logarithms ::对数的反对数属性

Example 10 ::例10

Example 11 ::例例11

Example 12 ::例例12

Example 13 ::例13

Example 14 ::例14

Summary ::摘要

Review ::回顾

Explore More ::探索更多

Answers for Review and Explore More Problems ::回顾和探讨更多问题的答复

PLIX ::PLIX

References ::参考参考资料

Logarithmic Properties
::对对数属性

Exponential Properties
::指数属性

Zero Exponent and Identity Properties
::零指数和身份属性

Zero Exponent and Identity Properties of Logarithms
::对数零指数和特性属性

Example 1
::例1

Example 2
::例2

Product Rule of Logarithms
::产品对数规则

Product Rule of Logarithms
::产品对数规则

Example 3
::例3

WARNING
::警告

Quotient Rule of Logarithms
::引引对数规则

Quotient Rule of Logarithms
::引引对数规则

Example 4
::例4

Example 5
::例5

WARNING
::警告

Power Rule of Logarithms
::对数规则的权力

Power Rule of Logarithms
::对数规则的权力

Example 6
::例6

Example 7
::例7

Example 8
::例8

WARNING
::警告

WARNING
::警告

Example 9
::例9

Inverse Functions
::反反函数

Inverse Property of Logarithms
::对数的反对数属性

Example 10
::例10

Example 11
::例例11

Example 12
::例例12

Example 13
::例13

Example 14
::例14

Summary
::摘要

Review
::回顾

Explore More
::探索更多

Answers for Review and Explore More Problems
::回顾和探讨更多问题的答复

PLIX
::PLIX

References
::参考参考资料